EIGEN VECTORS OF A MATRIX

Finding eigen vectors of a matrix is the  second step in solving a eigenvalue problem. To find eigenvectors, fist we have to find the eigenvalues from the given matrix. Then, we can find the eigenvectors that correspond to each eigenvalue found in the solution of the characteristic  equation. It is best to illustrate the procedure with an example.

Example :

Find the eigenvectors of

To find the eigenvectors, first we have to find the eigen values.

To find the eigen values, solve the following equation.

det|A - λI| = 0

where I is the 2 × 2 identity matrix.

The characteristic equation :

(5 - λ)(2 - λ) - 18 = 0

10 - 5λ - 2λ + λ2 - 18 = 0

λ- 7λ - 8 = 0

(λ - 8)(λ + 1) = 0

λ = 8  or  λ = -1

The two eigenvalues of the given matrix :

λ1 = 8

λ2 = -1

An eigenvector of a 2 × 2 matrix is going to be a column vector with 2 components. If we call these two unknowns x and y, then we can write the vector as

For the first eigenvalue, the eigenvector equation is

Av = λ1v

Specifically, we have the matrix equation,

From the above matrix equation, we have

5x + 2y = 8x

-3x + 2y = 0

3x - 2y = 0 ----(1)

9x + 2y = 8y

9x - 6y = 0

3x - 2y = 0 ----(2)

The linear equations (1) and (2) above represent the same line. So, the system has infinitely many solutions.

Solve (1) or (2) for y.

y = (³⁄₂)x

If we choose x = 2,

y = (³⁄₂)(2)

y = 3

Then the eigenvector corresponding to λ1 = 8 is

For the second eigenvalue, the eigenvector equation is

Av = λ2v

Specifically, we have the matrix equation,

From the above matrix equation, we have

5x + 2y = -x

6x + 2y = 0

3x + y = 0 ----(1)

9x + 2y = -y

9x + 3y = 0

3x + y = 0 ----(2)

The linear equations (1) and (2) above represent the same line. So, the system has infinitely many solutions.

Solve (1) or (2) for y.

y = -3x

If we choose x = 1,

y = -3(1)

y = -3

Then the eigenvector corresponding to λ1 = -1 is

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